I think one can enter a dispute regarding the notion of "accepted" but the idea is that General Relativity is successfully described by a Pseudo-Riemannian Manifold, subject to Einstein Equations, with free-falling objects following geodesics. Now you look for a set of axioms that give you this structure. One such set, although not entirely rigorous, is found in a paper by Ehlers, Pirani and Schild called the "The geometry of free fall and light propagation". I'll give you a brief discussion of the contents.
Start with two principles, (1) the Einstein Equivalence Principle and (2) the Finiteness of the velocity of light. The first one says that objects in free-fall in the gravitational field are in inertial movement and the second one says that not only light traves at a finite velocity but nothing goes faster than it.
Putting in another words principle (1) dictates that the trajectory of free-fall is as "straight" a line, from the point of view of an observer, as a constant velocity trajectory is in Newtonian physics. Principle (2) establishes that since everything travels at finite velocity there is a causal relation between events, namely if two events are have a spatial distance bigger than the velocity of light times the time separation they cannot have a cause-effect relationship, which of course implies in the relativity of simultaneity and other stuff from special relativity.
In more specific sense, principle (1) gives you a set of "straight lines", i.e. a set of geodesics, while principle (2) gives you a set of causal relations between events. In the paper by Ehlers, Pirani and Schild they call this two structures (1) a Projective Structure and (2) a Conformal Structure. Then they show that this two, with the assumptions that they are compatible and clocks behave in a reasonable way. imply the existence of a unique Lorentzian metric and Riemann tensor, together with the interpretation of geodesics and light-cones. All that's left if to require that the geodesics deviation be compatible with the one from Newtonian theory to obtain the Einstein Equations.
They do provide a set of axioms that address every part of the assumptions, but it all can be traced back to these two principles, Einstein Equivalence and finite velocity of the propagation of light.
As a remark, you may note that this idea is very different from what Weinberg exposes, namely that geometry is not fundamental in this description, but, as he himself says in page 147, this is an heterodox point of view not subscribed to by general relativists. On the other hand it is, as far as I'm aware, the mainstream point of view in String theory.